Some Observations on the 3x+1 Problem (original) (raw)
The 3x+1 problem: An annotated bibliography (1963--1999)
2003
The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts that for every positive integer n > 1 the forward orbit of n under iteration by T includes the integer 1. This paper is an annotated bibliography of work done on the 3x+1 problem and related problems from 1963 through 1999. At present the 3x+1 Conjecture remains unsolved.
The 3x+ 1 Problem: An Annotated Bibliography, II (2000-2009)
Arxiv preprint math/0608208, 2006
The 3x + 1 Conjecture asserts that each m ≥ 1 has some iterate T (k) (m) = 1. This is the second installment of an annotated bibliography of work done on the 3x + 1 problem and related problems, covering the period 2000 through 2009. At present the 3x + 1 Conjecture remains unsolved. This paper shows various results on the minimal elements having a given stopping time, where the "stopping time" is defined to be the number of odd elements in the trajectory up to and including 1. It also obtains a new congruential "sufficient set" criterion to verify the 3x + 1 Conjecture. It shows that knowing that the 3x + 1 Conjecture is true for all n ≡ 1 (mod 16) implies that it is true in general.
3x + 1 Problem Annotated Bibliography
The 3x + 1 problem concerns iteration of the map T : Z! Zgiven by T(x) = 3x + 1 2 if x 1 (mod 2) : x 2 if x 0 (mod 2) : The 3x + 1 Conjecture asserts that each m 1 has some iterate T (k) (m) = 1. This is an annotated bibliography of work done on the 3x + 1 problem and related problems subsequent to the survey of Lagarias (1985).
The 3x+1 Problem as a String Rewriting System
The 3x 1 problem can be viewed, starting with the binary form for any n ∈ N, as a string of "runs" of 1s and 0s, using methodology introduced by Błażewicz and Pettorossi in 1983. A simple system of two unary operators rewrites the length of each run, so that each new string represents the next odd integer on the 3x 1 path. This approach enables the conjecture to be recast as two assertions. I Every odd n ∈ N lies on a distinct 3x 1 trajectory between two Mersenne numbers 2 k − 1 or their equivalents, in the sense that every integer of the form 4m 1 with m being odd is equivalent to m because both yield the same successor. II If T r 2 k − 1 → 2 l − 1 for any r, k, l > 0, l < k; that is, the 3x 1 function expressed as a map of k's is monotonically decreasing, thereby ensuring that the function terminates for every integer.
The Structure of the 3x + 1 Function: An Introduction
1995
This section contains statements of all results that we have obtained to date. A few of the results are already known in the literature, but are included for ease of reference. The reader is encouraged to use the“Table of Symbols and Terms” on page 103 to look up definitions of terms, and, of course, in any of our papers, to use the Search facility that is available with all .pdf files on a web site. Proofs that are not given in this paper are given in the papers, “The Structure of the 3x + 1 Function”, “Are We Near a Solution to the 3x + 1 Problem?” and “A Solution to the 3x + 1 Problem” on the web site www.occampress.com. The term “[so]” following a lemma number means that the statement and proof of the lemma will be found in the paper, “A Solution to the 3x + 1 Problem” on the web site www.occampress.com. The term “[ar]” following a lemma number means that the statement and proof of the lemma will be found in the paper, “Are We Near a Solution to the 3x + 1 Problem?” on the web s...
arXiv (Cornell University), 2016
In this paper, we discuss the well known 3x + 1 conjecture in form of the accelerated Collatz function T defined on the positive odd integers. We present a sequence of quotient spaces and further, an invertible map, which are intrinsically related to the behavior of T. This approach allows to express the 3x + 1 conjecture in form of equivalent problems, which might be more accessible than the original conjecture.